Finite Group Databasebeta

← C22×C4⋊(C3⋊#157) #13181512

Subgroups — C22×C4⋊(C3⋊#157) #13181512

Order 1728.

1416 conjugacy classes of subgroups; only the 341 normal subgroups are shown.

normal subgroups only

This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 341 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3⋊#157C212yesmaximal
2C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
3C2×C4⋊(C3⋊#157)C2112yesmaximal, direct product factor
4C22×C4⋊#157—31nomaximal
5C23×C3×C3⋊D3C2211yescharacteristic
6C22×C3⋊#157—28no—
7C22×C3⋊#157C22112yes—
8C22×C33×C4C411yescharacteristic, complemented
9C22×C3×C3⋊Dic3C411yescharacteristic, complemented
10C2×C3×C4×C3⋊D3C416yescomplemented
11C2×C3×C4×C3⋊D3C2216yes—
12C4⋊(C3⋊#157)C22116yesdirect product factor
13C23×#157—32no—
14C22×C4×C3⋊D3D311yescharacteristic
15C22×C3×C4×D3—62no—
16C2×C4⋊#157—312no—
17C23×C33C2×C411yescharacteristic
18C22×C3×C3⋊D3C2×C413yes—
19C22×C3×C3⋊D3D414yes—
20C22×C3×C3⋊D3Q814yes—
21C22×C3×C3⋊D3C2313yes—
22C2×C3⋊#157—224no—
23C2×C3⋊#157C2318yes—
24C2×C33×C4C2×C416yescomplemented
25C2×C3×C3⋊Dic3C2×C416yescomplemented
26C3×C4×C3⋊D3C2×C4112yescomplemented
27C3×C4×C3⋊D3C2314yes—
28C22×C4⋊Dic3—91nomaximal
29C23×C3⋊D3C2×D311yescharacteristic
30C23×C3×D3—62no—
31C22×#157—312no—
32C22×#157—68no—
33C22×C32×C4—22no—
34C22×C32×C4—42no—
35C22×C32×C4Dic311yescharacteristic, complemented
36C22×C3⋊Dic3Dic311yescharacteristic, complemented
37C2×C4×C3⋊D3Dic316yescomplemented
38C2×C4×C3⋊D3C2×D316yes—
39C22×C3×Dic3—62no—
40C2×C3×C4×D3—612no—
41C2×C3×C4×D3—612no—
42C4⋊#157—316no—
43C22×C33C4⋊C414yescomplemented
44C22×C33C22×C413yes—
45C2×C3×C3⋊D3C4⋊C4112yescomplemented
46C2×C3×C3⋊D3C22×C413yes—
47C2×C3×C3⋊D3C2×D416yes—
48C2×C3×C3⋊D3C2×Q816yes—
49C2×C3×C3⋊D3C2411yescharacteristic
50C3⋊#157—28no—
51C33×C4C22×C414yescomplemented
52C3×C3⋊Dic3C22×C414yescomplemented
53C23×C3×C4—91no—
54C23×Dic3—92no—
55C22×C4×D3—62no—
56C2×C4⋊Dic3—912no—
57C23×C32—22no—
58C23×C32—42no—
59C23×C32C2×Dic311yescharacteristic
60C22×C3⋊D3Dic614yes—
61C22×C3⋊D3D1214yes—
62C22×C3⋊D3C2×Dic313yes—
63C22×C3⋊D3C22×D313yes—
64C22×C3×D3—66no—
65C22×C3×D3—68no—
66C22×C3×D3—68no—
67C22×C3×D3—66no—
68C2×#157—38no—
69C2×#157—624no—
70C2×C32×C4—212no—
71C2×C32×C4—412no—
72C2×C32×C4C2×Dic316yescomplemented
73C2×C3⋊Dic3C2×Dic316yescomplemented
74C4×C3⋊D3C2×Dic3112yescomplemented
75C4×C3⋊D3C22×D314yes—
76C2×C3×Dic3—612no—
77C3×C4×D3—624no—
78C3×C4×D3—68no—
79C22×C4⋊C4—271noSylow
80C2×C33C2×C4⋊C416yescomplemented
81C2×C33C23×C411yescharacteristic
82C3×C3⋊D3C2×C4⋊C416yescomplemented
83C3×C3⋊D3C22×D411yescharacteristic
84C3×C3⋊D3C22×Q811yescharacteristic
85C24×C3—91no—
86C23×D3—62no—
87C22×C3×C4—22no—
88C22×C3×C4—42no—
89C22×C3×C4—91no—
90C22×C3×C4—96no—
91C22×C3×C4—96no—
92C22×C3×C4#15711yescharacteristic, complemented
93C22×Dic3—62no—
94C22×Dic3—912no—
95C22×Dic3—188no—
96C2×C4×D3—612no—
97C2×C4×D3—612no—
98C4⋊Dic3—916no—
99C22×C32—26no—
100C22×C32—28no—
101C22×C32—48no—
102C22×C32—46no—
103C22×C32C4⋊Dic314yescomplemented
104C22×C32C22×Dic313yes—
105C2×C3⋊D3C4⋊Dic3112yescomplemented
106C2×C3⋊D3C2×Dic616yes—
107C2×C3⋊D3C2×D1216yes—
108C2×C3⋊D3C22×Dic313yes—
109C2×C3⋊D3C23×D311yescharacteristic
110C2×C3×D3—624no—
111C2×C3×D3—66no—
112C2×C3×D3—612no—
113C2×C3×D3—612no—
114C2×C3×D3—62no—
115#157—68no—
116C32×C4—28no—
117C32×C4—48no—
118C32×C4C22×Dic314yescomplemented
119C3⋊Dic3C22×Dic314yescomplemented
120C3×Dic3—68no—
121C23×C4—91no—
122C23×C4—272no—
123C2×C4⋊C4—2712no—
124C33C22×C4⋊C411yescharacteristic, complemented, Sylow
125C23×C3—22no—
126C23×C3—42no—
127C23×C3—93no—
128C23×C3—93no—
129C23×C3—94no—
130C23×C3—94no—
131C23×C3C2×#15711yescharacteristic
132C22×D3—68no—
133C22×D3—68no—
134C22×D3—66no—
135C22×D3—66no—
136C2×C3×C4—212no—
137C2×C3×C4—412no—
138C2×C3×C4—96no—
139C2×C3×C4—94no—
140C2×C3×C4—912no—
141C2×C3×C4C2×#15716yescomplemented
142C2×Dic3—612no—
143C2×Dic3—98no—
144C2×Dic3—1824no—
145C4×D3—624no—
146C4×D3—68no—
147C2×C32—212no—
148C2×C32—22no—
149C2×C32—412no—
150C2×C32—42no—
151C2×C32C2×C4⋊Dic316yescomplemented
152C2×C32C23×Dic311yescharacteristic
153C3⋊D3C2×C4⋊Dic316yescomplemented
154C3⋊D3C22×Dic611yescharacteristic
155C3⋊D3C22×D1211yescharacteristic
156C3×D3—612no—
157C3×D3—62no—
158C3×D3—62no—
159C24—91no—
160C22×C4—91no—
161C22×C4—96no—
162C22×C4—96no—
163C22×C4—2712no—
164C22×C4—548no—
165C22×C4C3⋊#15711yescharacteristic, complemented
166C4⋊C4—2716no—
167C22×C3—26no—
168C22×C3—28no—
169C22×C3—46no—
170C22×C3—48no—
171C22×C3—96no—
172C22×C3—96no—
173C22×C3—912no—
174C22×C3—91no—
175C22×C3—93no—
176C22×C3C4⋊#15714yescomplemented
177C22×C3C22×#15713yes—
178C2×D3—66no—
179C2×D3—62no—
180C2×D3—624no—
181C2×D3—612no—
182C2×D3—612no—
183C3×C4—28no—
184C3×C4—48no—
185C3×C4—94no—
186C3×C4C22×#15714yescomplemented
187Dic3—68no—
188Dic3—188no—
189C32—22no—
190C32—42no—
191C32C22×C4⋊Dic311yescharacteristic, complemented
192C23—93no—
193C23—93no—
194C23—94no—
195C23—94no—
196C23C2×C3⋊#15711yescharacteristic
197C2×C4—96no—
198C2×C4—94no—
199C2×C4—912no—
200C2×C4—278no—
201C2×C4—5424no—
202C2×C4C2×C3⋊#15716yescomplemented
203C2×C3—22no—
204C2×C3—212no—
205C2×C3—42no—
206C2×C3—412no—
207C2×C3—91no—
208C2×C3—96no—
209C2×C3—91no—
210C2×C3C2×C4⋊#15716yescomplemented
211C2×C3C23×#15711yescharacteristic
212D3—612no—
213D3—62no—
214D3—62no—
215C22—91no—
216C22—93no—
217C22—96no—
218C22—96no—
219C22—912no—
220C22C4⋊(C3⋊#157)14yesdirect product factor
221C22C22×C3⋊#15713yes—
222C4—94no—
223C4—548no—
224C4C22×C3⋊#15714yescomplemented
225C3—22no—
226C3—42no—
227C3C22×C4⋊#15711yescharacteristic, complemented
228C2—91no—
229C2—91no—
230C2—96no—
231C2C2×C4⋊(C3⋊#157)16yesdirect product factor
232C2C23×C3⋊#15711yescharacteristic
subgroupG/H
centerC23C2×C3⋊#157
commutatorC2×C33C23×C4
FrattiniC2C23×C3⋊#157
FittingC22×C33×C4C4
socleC23×C33C2×C4
prime radicalC23×C3×C3⋊D3C22
cosocleC2×C3×C3⋊D3C24
hypercenterC22×C4C3⋊#157
hypocenterC33C22×C4⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×C4⋊(C3⋊#157)C2direct (N × H)
8. C22×C33×C4C4semidirect
9. C22×C3×C3⋊Dic3C4semidirect
10. C2×C3×C4×C3⋊D3C4semidirect
12. C4⋊(C3⋊#157)C22direct (N × H)
24. C2×C33×C4C2×C4semidirect
25. C2×C3×C3⋊Dic3C2×C4semidirect
26. C3×C4×C3⋊D3C2×C4semidirect
35. C22×C32×C4Dic3semidirect
36. C22×C3⋊Dic3Dic3semidirect
37. C2×C4×C3⋊D3Dic3semidirect
43. C22×C33C4⋊C4semidirect
45. C2×C3×C3⋊D3C4⋊C4semidirect
51. C33×C4C22×C4semidirect
52. C3×C3⋊Dic3C22×C4semidirect
72. C2×C32×C4C2×Dic3semidirect
73. C2×C3⋊Dic3C2×Dic3semidirect
74. C4×C3⋊D3C2×Dic3semidirect
80. C2×C33C2×C4⋊C4semidirect
82. C3×C3⋊D3C2×C4⋊C4semidirect
92. C22×C3×C4#157semidirect
103. C22×C32C4⋊Dic3semidirect
105. C2×C3⋊D3C4⋊Dic3semidirect
118. C32×C4C22×Dic3semidirect
119. C3⋊Dic3C22×Dic3semidirect
124. C33C22×C4⋊C4semidirect
141. C2×C3×C4C2×#157semidirect
151. C2×C32C2×C4⋊Dic3semidirect
153. C3⋊D3C2×C4⋊Dic3semidirect
165. C22×C4C3⋊#157semidirect
176. C22×C3C4⋊#157semidirect
186. C3×C4C22×#157semidirect
191. C32C22×C4⋊Dic3semidirect
202. C2×C4C2×C3⋊#157semidirect
210. C2×C3C2×C4⋊#157semidirect
220. C22C4⋊(C3⋊#157)direct (N × H)
224. C4C22×C3⋊#157semidirect
227. C3C22×C4⋊#157semidirect
231. C2C2×C4⋊(C3⋊#157)direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×C3⋊#1572
C22×C3×C4×C3⋊D32
C2×C4⋊(C3⋊#157)2
C22×C4⋊#1573
C22×C4⋊Dic39